Let a and b be distinct positive real numbers. Consider the set X = z C : a|z|^2 + b Re (z) = a and b|z|^2 + a Re (z) = b . If the set X contains exactly two elements z₁ and z₂ , then the value of |z₁ - z₂|^2 is equal to

Step-by-step solution

Let z = x + iy , where x, y R . The given equations are: a(x^2 + y^2) + bx = a (1) b(x^2 + y^2) + ax = b (2) Multiplying equation (1) by b and equation (2) by a , we get: ab(x^2 + y^2) + b^2x = ab ab(x^2 + y^2) + a^2x = ab Subtracting the second equation from the first gives: (b^2 - a^2)x = 0 Since a and b are distinct positive real numbers, b^2 - a^2 0 . Therefore, x = 0 . Substituting x = 0 into equation (1), we get: a(0^2 + y^2) + 0 = a ay^2 = a y^2 = 1 Thus, y = 1 or y = -1 . The two elements in the set X are z